An Introduction
July 28, 2025
A Bayesian Framework for Persistent Homology(Maroulas, Nasrin, and Oballe 2020)
Fundamental geometric structures
A Simplicial Complex \(S\) is a collection of simplices satisfying:
| Bayesian framework for RVs | Bayesian framework for random PDs | |
|---|---|---|
| Prior | Modeled by a prior density \(f\) | Modeled by a Poisson PP with prior intensity \(\lambda\) |
| Likelihood | Depends on observed data | Stochastic kernel \(\color{purple}{\ell(y|x)}\) depends on observed PDs |
| Posterior | Compute the posterior density | Defines a Poisson PP with posterior intensity |
\[ \lambda_{\mathcal{D}_X | D_{Y^{1:m}}}(x) = \underbrace{\color{red}{(1 - \alpha(x))\lambda_{\mathcal{D}_X}(x)}}_{\text{Prior Vanished Part}} + \underbrace{\frac{1}{m} \alpha(x) \sum_{i=1}^m \sum_{y \in D_{Y^i}} \frac{ \color{purple}{\ell(y|x)}\color{blue}{ \lambda_{\mathcal{D}_X}(x)}}{\color{darkgreen}{\lambda_{\mathcal{D}_{Y_S}}(y)} + \color{blue}{\int_{\mathbb{W}} \ell(y|u) \alpha(u) \lambda_{\mathcal{D}_X}(u) du}}}_{\text{Update from Observed Points } y} \quad \text{a.s.} \]
| Case | Data Noise Std Dev (σ_Data) |
|---|---|
| Case I | 0.032 |
| Case II | 0.100 |
| Case III | 0.316 |
Problem:
Approach using Persistent Homology (PH):
Experiment Setup:
Results (AUC):
| Prior | Mean AUC | 5th Percentile | 95th Percentile |
|---|---|---|---|
| Prior-1 (Class-Specific) | 0.941 | 0.931 | 0.958 |
| Prior-2 (Common Flat) | 0.940 | 0.928 | 0.951 |
Conclusion:
The Bayesian framework achieves near-perfect classification accuracy (Mean AUC ≈ 0.94). Results are robust to the choice of prior (informative vs. flat). Demonstrates the framework’s capability for machine learning tasks on PDs for challenging real-world data.
Thomas Reinke